Monday, March 7, 2011

ONE ENTRY

Today, Form & Formalism brings you an entry on the notion of 'The One' in Badiou's work (focussing more or less exclusively on Being and Event). This text may or may not turn up in the BADIOU DICTIONARY, alongside the others I've been posting. An entry on 'The One', it turns out, had already been commissioned from another author (I'm not sure who), and so this entry is really just an understudy. I wasn't planning on preparing an entry on this concept at all, in fact, but just happened to stumble across it in a pile of scraps I had left over from my contribution to BADIOU: KEY CONCEPTS (the word limit on that text had been lowered at the last minute, and so I ended up with plenty of scraps on the cutting room floor---just as well, I suppose, as the short essay was starting to bristle with tangents). After a bit of pruning and scrubbing, the piece seemed to stand up fairly well as a dictionary entry, and so here it is:

========================

BADIOU DICTIONARY

ENTRY ON ‘THE ONE’

Z.L. FRASER

Being and Event begins with an announcement of the book’s inaugural decision: that ‘the one is not’ (23). There is no One, no self-sustaining unity, in being, but only the count-as-one, the non-self-sufficient operation of unification. There is no unity-in-itself, because every unity is a unity of something, something that differs from the operation of unification. This decision is no less fundamental for Badiou’s philosophy than his well-known equation of mathematics with ontology, and their metaontological meanings are deeply entangled; any attempt to isolate one from the other would mutilate the sense that Badiou gives its twin.

Through the prism of set theory, the non-being of the One refracts into three distinct ontological bans: (1) the prohibition of an ‘All’, or a set of all sets (by the ZF axioms, the supposition of such a set’s existence leads directly into the embrace of the Russell paradox)—an important corollary of this ban is that is no ‘set theoretical universe’ against which the various models of set theory can be measured, and so every coherent interpretation of the axioms will be pathological or ‘non-standard’ to some extent—there is no such thing as the standard model of set theory—a fact we must bear in mind when grappling with Cohen’s results, among others; (2) the ban on atomic elements, or units that are not themselves sets (ZF makes no provisions for unities that are not unities-of-something—with the possible exception of the void, into which the axiom of extensionality would collapse any putative ‘atoms’); and, we could add, (3) a self-unifying unity, a set that counts-as-one itself alone; schematically, the set W such that W = {W} (this set, which would in any case evade identification by the axiom of extensionality, is expressly forbidden by the axiom of foundation).

Note that there exist axiomatizations of set theory which violate each of the three impossibilities by which we have translated, ‘the One is not’. There are set theories with a universal set V, such that for all e, eÎ V (the One as All, as set of all sets);[i] there are set theories with urelements, elements u such that no element e belongs to u, but which are nevertheless distinct from the empty set (the One as atom, as a unity that is not a unity-of-something);[ii] and there are set theories with hypersets, sets X such that X Î X, or X Î A1Î … Î AnÎ X (the One as counting-itself-as-one, as self-presentation).[iii]

It is the identification of ontology not simply with mathematics but with a particular version of set theory (ZF) that therefore helps to motivate the decision that the One is not. Observing the existence of other One-affirming set theories emphasizes the particularity of this decision. The converse motivation—of the decision to identify mathematics with ontology by the decision on the non-being of the one—is somewhat murkier, but Badiou insists upon it. It is because the One is not, Badiou argues, that we must resist any temptation to subject being qua being to the unity of a concept. Subtracted from unity, ontology can articulate the sayable of being only by means of a non-conceptual regime of axioms, which regulate the construction of pure multiplicities without having recourse to any definition of multiplicity (BE 29). But why insist that concepts cannot deploy themselves axiomatically, by way of definitions that are purely implicit?[iv] Even if this is granted, nothing prevents a reversal of the argument. What, for instance, keeps an opponent from objecting to the placement of being-without-oneness under the ‘formal unity’ of an axiomatic, rather than submitting it to ‘the mobile multiplicity of the concept’?

[ii] See Ernst Zermelo, ‘Investigations in the Foundations of Set Theory I’, trans. Stefan Mengelberg, in Jean van Heijenoort, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Cambridge, Harvard University Press, 1967, pp. 183–198. Zermelo’s axiomatization of set theory with urelements, originally published in 1908, actually pre-dates the Zermelo-Fraenkel axiomatic which does without urelements and recognizes only sets as existent.

[iv] Replying to Frege’s accusation that an axiomatic operating with undefined terms is conceptually vacuous, the same Hilbert we met at the beginning of this chapter argues that, on the contrary,

If one is looking for other definitions of a ‘point’, e.g., through paraphrase in terms of extensionless, etc., then I must indeed oppose such attempts in the most decisive way; one is looking for something one can never find because there is nothing there; and everything gets lost and becomes vague and tangled and degenerates into a game of hide-and-seek. […T]o try to give a definition of a point in three lines [of text] is to my mind an impossibility, for only the whole structure of axioms yields a complete definition. Every axiom contributes something to the definition, and hence every new axiom changes the concept. (Hilbert, Letter to Frege, dated December 29th, 1899, pp. 39-40; e.a.)

It’s strange that Badiou would, without argument, reshackle the concept to the Fregean notion of definition, after arguing (some twenty years prior to writing Being and Event),

that the concepts of a science are necessarily of undefined words; that a definition is never anything more than the introduction of an abbreviating symbol; that, consequently, the regularity of the concept’s efficacy depends on the transparency of the code in which it figures, which is to say, on its virtual mathematization. (Badiou, ‘(Re)commencement de la materialisme dialectique,’ p.464, n.28; e.a.)

Do ‘concepts’ deployed in this fashion refasten discourse to the One? On what grounds could we even give an answer to this question?

Our series of entries for the BADIOU DICTIONARY will conclude tomorrow with THE VOID, so stay tuned!